Why use virtual displacement to make constraint forces vanish? Why do we use virtual displacement to vanish work done by constraint forces instead of the actual displacement?
 A: *

*In a nutshell, d'Alembert's principle states that a certain vector should be perpendicular to a constraint surface, i.e. perpendicular to all its tangent vectors, i.e. perpendicular to all infinitesimal virtual displacements. See also e.g.  this related Phys.SE post.


*Note that at this stage, we are just trying to find the EOMs of the physical system under investigation. At a later stage we will then try to solve the EOMs. We cannot use the actual displacement because we don't know the solution yet.


*Btw, it should probably be stressed that the actual displacement never coincides with a virtual displacement, because the latter is frozen in time, cf. e.g. this & this related Phys.SE posts.
A: H. Goldstein, Classical mechanics, Chapter $1$ says

"Note that if a particle is constrained to a surface that is itself moving in time, the force of constraint is instantaneously perpendicular to the surface and the work during a virtual displacement is still zero even though the work during an actual displacement in the time $dt$ does not necessarily vanish."

I think that this is just an additive advantage of taking virtual displacement instead of actual displacement and that the main reason leads to the linear independence of the equation
$$\sum_j \left\{\left[\frac{d}{dt}\left(\frac{\partial T}{\partial \dot q_j}\right)-\frac{\partial T}{\partial q_j}\right]-Q_j\right\}\delta q_j=0\tag{1.52}$$
which leads to the Lagrange's equations.
I used actual displacement $$dr_i=\sum_j \frac{\partial r_i}{\partial q_j}dq_j+\frac{\partial r_i}{\partial t}dt$$ instead of virtual displacement $$\delta r_j=\sum_j \frac{\partial r_i}{\partial q_j}\delta q_j\tag{1.47}$$ in the whole derivation of the Lagrange's equations and got an extra term depending on $\displaystyle\frac{\partial r_i}{\partial t}$ on the LHS of the equation $(1.52)$ which did not let me apply the logic of linear independence.
That was two years ago. I can't find that notebook and I am too lazy to do it again. You try it yourself. Comment below, should you face any issues.
PS: Again, this is not from a verified source.
